论文标题
Wigner $ d $ functions the Sphere的压缩感测
Compressive Sensing with Wigner $D$-functions on Subsets of the Sphere
论文作者
论文摘要
在本文中，我们证明了旋转组的限制测量域，$ \ mathrm {so}（3）$。我们首先在旋转组的测量子域$ r $ $ \ mathrm {so}（3）$上定义SLEPIAN函数。然后，我们从测量基础上将反问题转换为$ \ mathrm {so}（3）$的带限制的wigner $ d $ functions的有界正顺序系统，以限制信号稀疏性的方式，以一种限制的方式。使用Wigner $ d $ functions的对比方法，需要对所有$ \ mathrm {so}（3）$进行测量，我们表明SLEPIAN函数的正交性结构仅需要对sub-domain $ r $的测量，这是可选的。由于这种方法的细节以及$ r $浓度较低的SLEPIAN功能的固有存在，当研究信号良好地集中在$ r $上时，我们的方法具有最高的精度。与其他经典和压缩传感方法相比，我们提供了我们方法的数值示例。在重建质量方面，我们发现我们的方法优于我们测试的其他压缩传感方法，并且至少与经典方法一样好，但测量次数显着减少。
In this paper, we prove a compressive sensing guarantee for restricted measurement domains on the rotation group, $\mathrm{SO}(3)$. We do so by first defining Slepian functions on a measurement sub-domain $R$ of the rotation group $\mathrm{SO}(3)$. Then, we transform the inverse problem from the measurement basis, the bounded orthonormal system of band-limited Wigner $D$-functions on $\mathrm{SO}(3)$, to the Slepian functions in a way that limits increases to signal sparsity. Contrasting methods using Wigner $D$-functions that require measurements on all of $\mathrm{SO}(3)$, we show that the orthogonality structure of the Slepian functions only requires measurements on the sub-domain $R$, which is select-able. Due to the particulars of this approach and the inherent presence of Slepian functions with low concentrations on $R$, our approach gives the highest accuracy when the signal under study is well concentrated on $R$. We provide numerical examples of our method in comparison with other classical and compressive sensing approaches. In terms of reconstruction quality, we find that our method outperforms the other compressive sensing approaches we test and is at least as good as classical approaches but with a significant reduction in the number of measurements.
